In a cylinder with a piston, there are 0.05 m^3 of a gas (ideal) and the pressure is 1 bar. The piston is weightless and is held in place by a spring that follows Hooke's Law. At first, the spring just touches the piston and applies no force on it. The gas is heated until its volume increases by 60% and its pressure becomes 1.9 bar (the spring is outside the space where the gas is and gets compressed when the piston rises)
a) Find the equation that links the pressure with the volume of the gas, with the pressure in Pascals and the volume in cubic meters.
b) Calculate the work that the gas produces. What percentage of that work is consumed to compress the spring?
Given: surrounding environment pressure 1 bar
[Please help! I've been trying for a week! Please tell me how you did it, I want to understand the process.]
Answers & Comments
Verified answer
Let h be the initial height of the piston (in the original equilibrium state).
Let x be any additional distance the piston moves.
At any value of x:
V = A*(h+x)
Forces acting on the piston:
PA = kx + Patm*A
Also PV/T = constant = C
New V = 1.6*Ah
New P = 1.9*Patm
1.9*Patm*A = kx + Patm*A
kx = 0.9*Patm*A
We know x = 0.6*h
Work done on spring = 1/2 kx^2
= 0.45*Patm*A * 0.6h = 0.27*Patm*Ah
= 0.27*Patm*Vinit
Work done on surroundings = Patm*A*x
= Patm*ΔV = 0.9*Patm*Vinit
Work done by gas = sum of above
= 1.17*Patm*Vinit
= 1.17*1e5 * 0.05 = 5.85 kJ
% used to compress spring = 0.27/1.17 = 23%